Question: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}2x+6y &= 7 \\ -5x+3y &= 5\end{align*}$
Solution: Begin by moving the $y$ -term in the second equation to the right side of the equation. $-5x = -3y+5$ Divide both sides by $-5$ to isolate $x$ $x = {\dfrac{3}{5}y - 1}$ Substitute this expression for $x$ in the first equation. $2({\dfrac{3}{5}y - 1}) + 6y = 7$ $\dfrac{6}{5}y - 2 + 6y = 7$ Simplify by combining terms, then solve for $y$ $\dfrac{36}{5}y - 2 = 7$ $\dfrac{36}{5}y = 9$ $y = \dfrac{5}{4}$ Substitute $\dfrac{5}{4}$ for $y$ in the top equation. $2x+6( \dfrac{5}{4}) = 7$ $2x+\dfrac{15}{2} = 7$ $2x = -\dfrac{1}{2}$ $x = -\dfrac{1}{4}$ The solution is $\enspace x = -\dfrac{1}{4}, \enspace y = \dfrac{5}{4}$.